Question

Point C is located at (1, 2) and point D is located at (−4, −2). Find the x value for the point that is 1 over 4 the distance from point C to point D. (5 points)

Answer 1

Check the picture below.

$\textit{internal division of a segment using a fraction}\\\\ C(\stackrel{x_1}{1}~,~\stackrel{y_1}{2})\qquad D(\stackrel{x_2}{-4}~,~\stackrel{y_2}{-2})~\hspace{8em} \frac{1}{4}\textit{ of the way from }C\to D \\\\[-0.35em] ~\dotfill\\\\ \stackrel{~\hfill \textit{component form of segment CD}}{ (\stackrel{x_2}{-4}-\stackrel{x_1}{1}, \stackrel{y_2}{-2}-\stackrel{y_1}{2})\qquad \implies \qquad (-5~~,~~-4)} \\\\[-0.35em] ~\dotfill$

$\stackrel{~\hfill \textit{values to be added to Point C coordinates}} {\cfrac{1}{4}(-5~~,~~-4)\implies \cfrac{1}{4}(-5)~,~\cfrac{1}{4}(-4)\qquad \implies \qquad \left(-\frac{5}{4}~~,~~-1\right)} \\\\\\ \stackrel{\textit{addition of Point C coordinates and values}~\hfill } {\left(-\frac{5}{4}~~,~~-1\right)+\left( 1~~,~~2\right) \implies \left(-\frac{5}{4}+1~~,~~-1+2 \right)}\implies \underset{\stackrel{\uparrow }{x}\qquad }{\left(-\frac{1}{4}~~,~~1 \right)}$